Arithmetic, Geometric and Other Types of AveragesFebruary 2019

ARITHMETIC, GEOMETRIC AND OTHER TYPES OF AVERAGES

JANIS ZVINGELIS

1. Types of Average Returns

There are many types of averages used in the applied finance today. There are arithmeticand geometric averages; realized and forecasted averages; real and nominal averages, andothers. All of these different types of averages are routinely used in the finance industry(usually when applied to returns), but often without referencing the type of average that isused, which, more often than not, leads to confusion. In this note I will define, compare andgives examples on how and when to use which type of average. Finally, I will also cover severalways of annualizing the average returns.

2. Arithmetic vs Geometric Average Return

Let’s start with the two best known and widely used types of averages: arithmetic andgeometric averages.

2.1. Arithmetic. The popularity of the arithmetic average can be, in part, explained byits simplicity. Let’s use rt to denote the rate of return during the time period t. Then thearithmetic average rA over the investment horizon with T periods is calculated as follows

rA ≡∑T

t=1 rtT

(2.1)

The arithmetic average has a very nice property: the longer performance sample oneuses to estimate the average, the more “confidence” one can have in the obtained result. Toexplore this property, I have calculated the arithmetic average return series for the (real) S&P500 series from January of 1871 to present. In particular, in Figure 1 I have plotted thedistributions of averages from non-overlapping time series, where the time series being usedin forming the averages vary in their length. Thus, I am comparing the set of all arithmeticaverage returns over non-overlapping three-, 24-, and 120-month periods. While the overallaverage across these series is exactly the same (about 0.27 percent of real price return permonth), the dispersion of these average returns is vastly different. The most important pointto note is that the longer the data series used calculating the arithmetic average, the more

QRG, Envestnet, 35 East Wacker Drive, Suite 2400, Chicago, IL, 60601Date: November 27, 2019.

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concentrated around the overall mean the dispersion of the averages becomes. This phenom-enon in statistics is called the “Law of Large Numbers (LLN)”1, and it is, along with thesimplicity of arithmetic average, the main reason for its popularity.

Figure 1. Plot of the distribution of the arithmetic average returns over various (non-overlapping) horizons for S&P 500 (1871/01 - 2017/09, real price returns in decimal format).Source: www.multpl.com & QRG.

Another application of arithmetic mean return is in forecasting. It turns out that thearithmetic average return can be used as a good predictor (due to the Law of Large Numbersthat we mentioned earlier) of the so-called “expected return”. “Expected return” can bethought of as an unknown average that we are trying to forecast. However, unlike an arithmeticaverage, where each observation is weighed by one divided by the number of total observations(see equation 2.1), the “expected value” construction uses the likelihood of a particular valuehappening as its weight.2 Because of the property of the arithmetic average is a good predictor

1This property (i.e., the variance of the average shrinking, when there are more terms included in theaverage), when applied to returns, is sometimes mistakenly called the “time diversification”. This mistakenargument then is used to justify why holding equity investments make sense, especially over the long term.The argument goes as follows: the longer the holding period (i.e., the more terms there are in the arithmeticaverage), the lower the variance/volatility of the investment. Hence, investing in higher volatility asset classes(such as equities) makes sense, especially over longer investment horizons. The problem with this argument isthat the variance/volatility of the arithmetic mean return does indeed shrink (due to LLN), but investors areusually not interested with the variances/volatilities of averages of returns, but rather with variances/volatilitiesof realized return series during some time period. A more convincing argument behind “time diversification”(i.e., the idea for why holding equities over longer term rather shorter term make sense, was advanced byCampbell & Viceira (2005). It relies on the mean-reversion exhibited by equity returns through time.

2There is a lot of confusion regarding the concept of the “expected value” or “expected return”, mostlydue to the difference in the usage of the word “expected” in statistics and in everyday language. In statistics,

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of the “expected return” (the larger the sample used to construct the arithmetic average return,the closer it is to the unknown “expected return”), the terms “arithmetic average return” and“expected return” are used interchangeably, especially in the applied finance.

Finally, while the arithmetic mean return is often used in summarizing a performanceof an asset through time, it can result in misleading results, especially during high volatilityperiods. Table 1 gives a stylized example of a two-period example, where the starting amountand ending amount are the same ($100), but the arithmetic return is positive 25 percent.Clearly, this is a misleading result and points to one of arithmetic return’s major weaknesses– inability to correctly summarize performance in high volatility environments. Another typeof average return – the geometric average return – does a much better job in these situations,and we now turn to a closer look at this average.

End-of-Period PeriodPeriod Balance Return

0 $1001 $50 -50%2 $100 100%

arithm.avg. 25%3

geom.avg. 0%4

Table 1. Performance summary with arithmetic and geometric average returns.

2.2. Geometric. Let’s give a formula that defines the geometric average return. Let’s use rtto denote the rate of return during the time period t. Then the geometric average rG over theinvestment horizon with T periods is calculated as follows

rG ≡

(T∏t=1

(1 + rt)

)1/T

− 1(2.2)

the word “expected” (as in “expected return”) refers to a kind of arithmetic average that we just described.The word “expected” in everyday language is used with a connotation of “likely to happen” or “forecasted”.As we will see in the next section, it turns out that an “expected wealth”, for example, certainly cannot beexpected to happen. Because people in finance are not always aware of this distinction, one often hears suchphrases as “expected average return”, which in statistics would mean (because “expected” again means a kindof an average) an “average average return” – certainly not what the user intended. A better term to use infinance would be “forecasted average return”. Importantly, various types of different approaches, besides thearithmetic average return (and its close cousin “expected return”), can be used to construct the forecastedaverage return. It is therefore important to keep the distinction of “expected return” and “forecasted averagereturn” in mind.

3(−50% + 100%)/2 = 25%.4(100/100)1/2 − 1 = 0%. That is, ending balance divided by the beginning balance, all raised to the power

of one over the number of holding periods, minus one.

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While not as popular as the arithmetic average return,5 geometric return has several veryuseful properties.

First, as demonstrated in table 1, geometric average return is unaffected by the volatilityin the return, and therefore accurately captures performance through time. It therefore shouldbe the go-to measure of summarizing financial performance (e.g., manager, account, etc)through time. In fact, the following approximate relationship (see appendix for the proof)holds between the arithmetic and geometric rates of return:

rG u rA − 0.5V (rt).(2.3)

What it says is what we observed in table 1: the higher the volatility of returns, the fartheraway the geometric average return will be from the arithmetic average return. Also, geometricreturn will always be no smaller than arithmetic return, and for cases of very small volatility(e.g., cash), the arithmetic and geometric average returns are essentially the same.

Second, geometric return is very useful in forecasting the likely future value of cumulativereturn or wealth, WT , defined as WT =

∏Tt=1 (1 + rt), where the sequence of rt’s, t = 1, . . . , T ,

represents the random sequence of future returns. A common estimator of the future wealth,WT , is the expected wealth (more on this in the next section), which can be forecasted bythe following estimator:6 It turns out that this estimator is consistently too high. Figure 2shows long history of (real) price series (so, cumulative wealth) of S&P 500, along with twopredictors of this series, one of which is based on arithmetic average return. Note how thisestimator gets ever farther away (on the upside) from the series its trying to predict, as theforecast horizon increases. This highlights the problem of using the arithmetic return in theformula (1 + rA)T to predict WT .7

It turns out that a much better predictor of future values of wealth/cumulative return,WT is based on geometric average return: (1 + rG)T . As demonstrated in figure 2, thisestimator is consistently in the ballpark of the WT value through time. Appendix also givesa proof that the estimator of future wealth, based of geometric average return ((1 + rG)T ),predicts the median value of WT , and, thus, is more appropriate as a predictor, especially forlong investment and prediction horizons.

Finally, a note of caution. I sometimes come across a notion that using geometric averagereturns in simulations is somehow “more conservative” and therefore presumably better. Thisis misguided on two fronts. First, as the equation 2.3 demonstrates, arithmetic and geometricaverage returns are related to each other in certain mathematical ways. For this reason theyshould be thought of as representing the same level of average return, with the only differencethat they are expressed in different units. The same way as a 1,000 Japanese Yen burger doesnot become cheaper when translated into dollars (about 9.22 US Dollars at the time of writingthis), the average return does not become lower when translated from arithmetic average returnto geometric return – just the units are different. Second, behind most simulation engines is

5The main reason for this is that the geometric average return does not possess certain nice statisticalproperties (to be discussed in the next section), mostly because it is a product, rather than a sum, of returns.Products of returns are harder to handle, from a statistical point of view, than sum of returns.

6E(WT ) = E(∏T

t=1 (1 + rt)) =∏T

t=1 (1 + E(rt)), and an estimator for E(rt) is the arithmetic average rA.

Thus, the estimator for WT is (1 + rA)T .7Appendix also gives a mathematical proof of this and shows that the probability of the (1+ rA)T exceeding

WT approaches one (certainty) as the investment horizon increases.

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Figure 2. Plot of the (real) price series of S&P 500 (1871/01 - 2017/09, in logs). Thefigure also contains two cumulative return series – one aggregated at the average arithmeticreturn (about 0.27% monthly real return) and the other at average geometric return (about0.19% monthly real return), where the averages are calculated over the time period 1871/01- 2017/09. The starting value of the cumulative return series is the price level of S&P 500 in1871/01. Source: www.multpl.com & QRG.

a statistical concept of “density function” – a way to assign probability to various levels ofreturns. One of the inputs to these “density functions” is always in terms of arithmetic averagereturn. Thus, it would be a mistake to enter the geometric average return as an input intoreturn simulators, because they are most likely expecting an arithmetic average return input.As a final point of advice – if a more conservative (lower) average return is desired as an inputto the simulation engine, it is preferable to reduce the arithmetic average return as needed,while being cognizant of making this reduction, rather than switching to geometric averagereturn units in a logically unsound effort to be “more conservative” with one’s capital marketestimates.

3. Nominal vs Real Average Returns

In this section we will discuss the inflation adjustment applied to returns. I often comesacross the concept of “inflation adjusted” returns and usually it puzzles me, because I cannotfigure out whether the inflation adjustment is converting future/past values to today’s dollarsor the other way around.

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A much clearer approach is to calculate returns in either “real” (i.e., today’s dollar) termsor “nominal” terms, where the latter refers to the return without any inflation adjustment.Based on this we can define a “nominal average return” (either arithmetic or geometric), rnA orrnG, which refers to the arithmetic or geometric return in the units that it is observed, withoutany inflation adjustment. Also, we can define a “real average return” (either arithmetic orgeometric), rrA or rrG, which refers to the arithmetic or geometric return that takes the nominalreturn and factors out the inflation. Thus, if inflation is positive, as it usually is, the real returnwill be smaller than the nominal return.

rn u rr + i,(3.1)

where i refers to the rate of annual inflation.There are only a few financial parameters that can be observed in real terms, with most

being quoted in nominal terms. Two that I am aware of are the yields on Treasury InflationProtected Securities (TIPS) and dividend yields paid out by equities.

4. Annualization

In this section I will briefly cover the topic of annualization. Oftentimes, the rates ofreturn or yields or other quantities (e.g., Sharpe ratios and volatilities) are given in the unitsunderlying the calculations. For example, if I observe historical monthly nominal returns onS&P 500 index and calculate an arithmetic average return from these returns, the arithmeticaverage return will be given in monthly units. However, what if I want to use my calculatedarithmetic average return (in monthly units) to investigate whether year 2018 was an aboveor below performance year for equities? To carry out this comparison, I will need to annualizemy monthly arithmetic average return.

There are generally two types of approaches that can be used to annualize an average re-turn: linear approach (based on the concept of geometric Brownian motion)8 or compoundingapproach.

The linear approach to annualizing a return says that if we observe/caculate a return inunits that are less than a year (say, months), then to get to an annual quantity we need tomultiply the original return by the number of time units that fit in a year. For example, if wecalculate the arithmetic average return based on monthly returns and then want to annualizeit, we should multiply this average by twelve.

The compounding approach, which is better applicable to the geometric average returnis given by the following formula:

rG,a = (1 + rG,h)N − 1,

where rG,a and rG,h refer to annual and higher frequency (e.g., monthly) geometric averagerates of return, respectively, and N refers to the number of times that the higher observationalfrequency fits into a year.

8The assumption that a price series follows a geometric Brownian motion is used to obtain the fa-mous Black-Scholes option pricing model. The geometric Brownian motion assumption also implies that

log(∏T

t=1(1 + rt))∼ N(µT, σ2T ), where rt is the return observed during a certain time period.

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In addition to annualizing returns, we can also annualize standard deviations, variances,covariances, Sharpe ratios and alike. This task is slightly more complicated task, but we willbe using the geometric Brownian motion assumption that we referenced above. That is, we

will be using the assumption that log(∏T

t=1(1 + rt))∼ N(µT, σ2T ). In particular, we will use

the assumption that variance of a compounded return log return log(∏T

t=1(1 + rt))

is equal

to σ2T . That is, the variance is linear in time. Thus, if we calculate a variance that is basedon monthly returns and want to annualized it, we would multiply it by 12 – the number ofmonths in a year. Note that the comparable approach to standard deviation/volatility wouldbe to multiply the monthly standard deviation by a square root of 12.

Furthermore, when annualizing volatility, especially in the context of calculating Sharperatios, we need to be mindful of serial correlation in returns (e.g., negative serial correlationis observed as mean-reversion in returns). This issue is highlighted in Lo (2002), who pointsout that in the absence of accounting for the serial correlation in volatility calculations, theSharpe ratio is increasing in the units of standardization. That is, suppose that you calculateda daily return and volatility of a trading strategy and then proceeded to calculate its Sharperatio based on these daily values. If you then proceeded to convert this daily Sharpe ratioto a monthly Sharpe ratio or even an annual Sharpe ratio, using the approaches we reviewedabove, then the monthly Sharpe ratio would be larger in absolute value than the daily Sharperatio, with the annual Sharpe ratio larger still.9

5. Application on the Platform

In this section we briefly touch upon the Capital Market Assumptions (CMAs) that areassociated with Envestnet’s platform. The CMAs that are loaded unto Envestnet’s platformare given in annual nominal arithmetic average units. Also, the simulation engine that drivesteh proposal expects that the CMAs be presented in annual nominal arithmetic average units.However, as described above, if needed, we can easily translate average returns given in otherunits (e.g., monthly real geometric average) to those in annual nominal arithmetic averageunits. As in the above sections, let’s denote the nominal geometric average return by rnG;real geometric return by rrG; nominal arithmetic average return by rnA; real arithmetic averagereturn by rrA. Then the flowchart below summarizes the relationships given above:

rnAOOeq. 2.3

��

ooeq. 3.1 // rrAOOeq. 2.3

��rnG

ooeq. 3.1

// rrG

9The reason for this is that the numerator (the average return) of the Sharpe ratios is multiplied by thenumber of periods in a year, while the denominator (the volatility) is multiplied by the number of periods ina year squared. Thus, the original Sharpe ratio (say, daily) is multiplied by a square of number of units oforiginal observation that fit into the period to which you want to standardize the Sharpe ratio (e.g., 20 businessdays in a month; about 63 business days in a quarter, and about 252 business days in a year). If we don’taccount serial correlation, in a Sharpe ratio annualization exercise the denominator always grows slower thanthe numerator, when carrying out the annualization, which leads to misleading Sharpe ratio estimates.

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For example, to translate the nominal arithmetic average return, rnA, into real geometricaverage return, rrG, all we need to do is apply equations 2.3 and 3.1. While the order in whichthese translating equations (arithmetic to geometric vs nominal to real) are applied matters, inprinciple, one can safely apply them disregarding the order, as the error in this approximationwill not be meaningful.

6. Appendix

6.1. Arithmetic-to-Geometric Translation Formula. Let’s assume that a gross randomreturn, 1 + rt, has a log-normal distribution for all t = 1, . . . , T . What this means is that thenatural log of 1 + rt is distributed as a normal random variable with parameters µ and σ:log(1 + rt) ∼ N(µ, σ2), which then results in the following formulas for the expected (gross)returns, E(1 + rt), and variances, V (1 + rt):

E(1 + rt) = eµ+σ2/2(6.1)

V (1 + rt) =(eσ

2 − 1)· e2µ+σ2

(6.2)

Solving for µ and σ in the above system of equations we obtain the following solutions (see,for example, de La Grandville (1998)):

µ = log(1 + E(rt))− 0.5 log

(1 +

V (1 + rt)

(1 + E(rt))2

)(6.3)

σ2 = log

(1 +

V (1 + rt)

(1 + E(rt))2

)(6.4)

Now, let’s derive the relationship between the geometric average return and the arithmeticaverage return:

1 + rG =

(T∏t=1

(1 + rt)

)1/T

(by definition of rG in eq. 2.2)

= elog((∏T

t (1+rt))1/T)

= e∑T

t log(1+rt)/T

u eE(log(1+rt)) (by the Law of Large Numbers)

= eµ (by an assumption log(1 + rt) ∼ N(µ, σ2))

= elog(1+E(rt) · e−0.5σ2(by eq. 6.3 & 6.4)

u (1 + E(rt)) · (1− 0.5σ2) (by Taylor series: ex = 1 + x+ x2/2! . . .)

u 1 + E(rt)− 0.5σ2

u 1 + E(rt)− 0.5V (rt) (by eq. 6.4 and

Taylor series: log(1 + x) = x− x2/2− . . .)u 1 + rA − 0.5V (rt) (by the Law of Large Numbers)

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Therefore, we obtain the following conversion formula between the arithmetic and geometricmean returns:

rG u rA − 0.5V (rt)(6.5)

Note that this formula is an approximation and holds best, when A. the number of elementsin the average is large; B. individual returns tend to be Normally distributed; C. σ, E(rt), andV (rt) are relatively close to zero. In most cases, these assumptions hold, at least partially,and the approximation in equation 6.5 works very well.

6.2. Predicting Wealth with Expected Wealth. In this section we will demonstrate thatthe probability of cumulative return (or wealth) exceeding the expected value of wealth/cumulativereturns tends to zero as the investment horizon increases. Also, see sections above as well asfigure 2 for a plot supporting this claim.

Let’s define WT to be the cumulative return/wealth at time T from investing one dollar T

periods ago: WT =∏Tt=1 (1 + rt). Wealth, WT , when standing at time zero, is a random vari-

able, since the sequence of future returns (rt, t = 1, . . . , T ) is random. Oftentimes, researchersare interested in predicting the value of WT , and an estimator, which looks very reasonableon the face of it, is the expected wealth:

E(WT ) = E

(T∏t=1

(1 + rt)

)

=

T∏t=1

(1 + E(rt)) (by IID assumption of rt’s)

This estimator seems reasonable, because as we mentioned in previous sections E(rt) is thebest (in some certain mathematical sense) predictor of rt. It therefore may stand to reasonthat the expected wealth might be a good predictor for WT . However, as it turns out, expectedwealth is not a good predictor of WT , and, in fact, the probability of the expected wealth,E(WT ), exceeding the realized wealth, WT , converges to 1 (certainty) as the investment horizonincreases to infinity. In other words, if the investment horizon is long, the realized wealth willalmost always fall below the forecasted wealth number of expected wealth.

Why such a seemingly nonintuitive result? What is happening is that WT is a randomvariable that is constrained from below by zero (i.e., can’t lose more than what’s invested),which means that WT is a skewed distribution, in this case skewed to the positive side. Thismeans that the expected value of this distribution, E(WT ), will always be no smaller (and, infact, much larger, the more skewed the distribution, i.e., the longer the investment horizon)than the median of this distribution.

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Next, let’s prove the claim that the probability of expected wealth exceeding the wealthtends to zero with certainty as the investment horizon grows:

P (Wt ≥ E(WT )) = P

(T∏t=1

(1 + rt) ≥ E

(T∏t=1

(1 + rt)

))(by definition of WT )

= P

((e∑T

t=1 log(1+rt)/T)T≥(elogE(1+rt)

)T)(by IID assumption on rt’s)

= P

(T∑t=1

log(1 + rt)/T ≥ logE(1 + rt)

)

= P

(T∑t=1

log(1 + rt)/T − E (log(1 + rt)) ≥

logE(1 + rt)− E (log(1 + rt))︸ ︷︷ ︸C

( C > 0 by Jensen’s inequality)

≤ P

(|T∑t=1

log(1 + rt)/T − E (log(1 + rt)) | ≥ C

)

≤ 1

C2E

(|T∑t=1

log(1 + rt)/T − E (log(1 + rt)) |2)

by Chebychef’s inequality)

=1

C2V ar

(T∑t=1

log(1 + rt)/T

)

=1

C2V ar (log(1 + rt)) /T (by IID assumption on rt’s)

T→∞−−−−→ 0

Note that, since the expected return, E(rt), is predicted by arithmetic mean return (byLLN), the above claim that the expected wealth is a poor predictor of future wealth alsoapplies to the epxpected wealth estimate, where E(rt) is replaced with the arithmetic meanreturn, rA.

6.3. Predicting Wealth with Median Wealth. In this section we explore the claim thata better predictor, compared to the expected wealth analyzed in previous section, of futurewealth might be the median wealth.10 Importantly, median wealth is related to the geometricmean, and we will demonstrate this relationship precisely in this section. Also, see sectionsabove as well as figure 2 for a plot supporting this claim.

Let’s assume that the return has a log-normal distribution, i.e., log(1 + Rt) ∼ N(µ, σ2).

Then the cumulative return/wealth (WT =∏Tt=1(1 + rt)) also has a log-normal distribution:11

10We use results from Hughson, Stutzer, & Yung (2006).11We are assuming the rt’s are IID.

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log(WT ) ∼ N(µT, σ2T ). Given this, we can write the expression for WT as follows:

WT =T∏t=1

(1 + rt)

= eµT+Z·σ·√T ,(6.6)

where Z ∼ N(0, 1). Given the above, it’s easily seen that the median wealth can be calculatedas follows:12

med(WT ) = eµT (set Z = 0, its median, in eq. 6.6)

u e(∑T

t=1 log(1+rt)/T)T (by LLN)

u

(T∏t=1

(1 + rt)

)T/Tu (1 + rG)T (by definition of geometric mean return in eq. 2.2)

Thus, median wealth (or cumulative return) is approximately equal to a geometric return thatis accumulated to the power equal to the investment periods.

References

Campbell, John Y., & Luis M. Viceira, 2005, The term structure of the risk-return tradeoff, Financial AnalystsJournal 61(1), 433–495.

de La Grandville, Olivier, 1998, The long-term expected rate of return: Seting it right, Financial AnalystsJournal 54:6, 75–80.

Hughson, Eric, Michael Stutzer, & Chris Yung, 2006, The misuse of expected returns, Financial AnalystsJournal 62(6), 88–96.

Lo, Andrew W., 2002, The statistics of Sharpe ratios, Financial Analysts Journal 36–52.

12Turns out that the following key expression for a median also holds approximately when the return is notassumed to follow a log-normal distribution (Hughson et al. 2006).